List of speakers

  • Nikolay Abrosimov (Sobolev Institute of Mathematics and Novosibirsk State University), The volume of a spherical antiprism with S2n symmetry

  • Paolo Cavicchioli (Università degli studi di Modena e Reggio Emilia), An algorithmic method to compute plat-like Markov moves for genus two 3-manifolds

  • Seonmi Choi (Kyungpook National Unversity), On algebraic structures related to skein relations and their invariants

  • Andrey Egorov (Novosibirsk State University), On volumes of hyperbolic right-angled polyhedra

  • Olga Frolkina (M.V. Lomonosov Moscow State University), A solution of J.Cobb's question

  • Sergei Gukov (California Institute of Technology), Non-semisimple TQFT's and BPS q-series

  • Maxim Ivanov (Novosibirsk State University), Recurrent construction of virtual link invariants arising from flat links

  • Suhyeon Jeong (Pusan National University), Pseudoknots and the Goeritz matrix

  • Louis H Kauffman (University of Illinois at Chicago), Invariants of Flat Virtual Knots and Links

  • Mikhail Khovanov (Columbia University), Introduction to universal construction and foam evaluation

  • Jieon Kim (Pusan National University), On symmetric biquandles and invariants for unoriented surface-links

  • Seongjeong Kim (Jilin University), On parities for knots in Sg×S¹

  • Seonhwa Kim (University of Seoul), Generalized Riley polynomials of a knot and trace fields of representations

  • Tatyana Kozlovskaya (Regional Mathematical Center, Tomsk State University), The singular pure braid group

  • Sergei Lando (Higher School of Economics, Skolkovo Institute of Science and Technology), Lie algebras weight systems and graph invariants

  • Vassily Olegovich Manturov (Moscow Institute of Physics and Technology), Three faces of sliceness for knots: pictures, groups, matchings
  • Sergey Melikhov (Steklov Mathematical Institute), Topological isotopy and Cochran's derived invariants

  • Sam Nelson (Claremont McKenna College), Biquandle Bracket Quivers

  • Vladimir Mikhajlovich Nezhinskij (St Petersburg State University, Herzen University), Spatial framed graphs and their isotopy

  • Igor Nikonov (Moscow State University), On the principle of indistinguishability for crossings of classical knots

  • Eiji Ogasa (Meiji Gakuin University), Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links

  • Jin Seok Oh (Kyungpook National Unversity), Torsion subgroups of homology of finite dihedral quandles

  • Mohd Ibrahim Sheikh (Pusan National University), Bikei’s, Biquandles and Dichormatic Links

  • Hongdae Yun (Kyungpook National University), The Betti numbers for the set-theoretic Yang-Baxter homology groups of Alexander biquandles

  • Bao Huu Vuong (Tomsk State University), On hyperelliptic Euclidean 3-manifolds

  • Seung Yeop Yang (Kyungpook National Unversity), Set-theoretic Yang-Baxter cohomology groups of finite cyclic biquandles

ABSTRACTS

Nikolay Abrosimov, Sobolev Institute of Mathematics
and Novosibirsk State University

The volume of a spherical antiprism with S2n symmetry.

   We consider a spherical antiprism. It is a convex polyhedron with 2n vertices in the spherical space S^3. This polyhedron has a group of symmetries S2n generated by a mirror-rotational symmetry of order 2n, i.e. rotation to the angle π/n followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedron in S^3. Then we find relations between its dihedral angles and edge lengths in the form of cosine rules through a property of a spherical isosceles trapezoid. Finally, we obtain an explicit integral formula for the volume of a spherical antiprism in terms of the edge lengths.  

Paolo Cavicchioli, Università degli studi di Modena e Reggio Emilia
An algorithmic method to compute plat-like Markov moves for genus two 3-manifolds

      The talk deals with equivalence of links in 3-manifolds of Heegaard genus 2. Starting from a description of such a manifold introduced in [1], that uses 6-tuples of integers and determines a Heegaard decomposition of the manifold, we construct an algorithm (implemented in c++) which allows to find the words in B_{2, 2n}, the braid group on 2n strands of a surface of genus 2, that realizes the plat-equivalence for links in that manifold. In this way we extend to the case of genus 2 the result obtained in [2] for genus 1 manifolds. We describe explicitly the words for a notable group of manifolds.

[1] Casali, M. R. & Grasselli, L. 2-symmetric crystallizations and 2-fold branched coverings of S3.Discrete Math. 87, 9–22 (1991)

[2] Cattabriga, A. & Gabrovšek, B. A Markov theorem for generalized plat decomposition. Ann.Sc. Norm. Super. Pisa Cl. Sci. XX, 1273–1294 (2018)

Seonmi Choi, Kyungpook National Unversity
On algebraic structures related to skein relations and their invariants

   Przytycki and Traczyk introduced a new algebraic structure, called the Conway algebra, and constructed invariants of oriented links valued in Conway algebras. Niebrzydowski and Przytycki defined a Kauffman bracket magma and constructed an invariant of framed links. These invariants are closely related to polynomial invariants. In this talk, we will define their generalizations for surface-links and construct invariants via marked graph diagrams. Moreover, we will define a specific map on a Kauffman bracket magma and construct some invariants for oriented links or oriented surface-links.

Andrey Egorov, Novosibirsk State University
On volumes of hyperbolic right-angled polyhedra

   In three-dimensional Lobachevsky space consider right-angled polyhedra. We will look at some properties of this polyhedra and consider new upper bounds on volumes of right-angled polyhedra in hyperbolic space in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite vertices, and for finite volume polyhedra with vertices of both types. In addition, we will look at some connections with knot theory.

Olga Frolkina, M.V. Lomonosov Moscow State University
A solution of J.Cobb's question

    Questions about the projections of zero-dimensional sets were considered already at the end of the 19th century. In 1884 G.Cantor  described the surjection of the middle-thirds Cantor set onto the segment [0,1]; its graph is zero-dimensional, but gives the unit segment when projecting on the Oy-axis. Cantor sets in plane all of whose projections are segments were constructed by L.Antoine (1924), H.Otto (1933), A.Flores (1933), G.Noebeling (1933). In 1947, K.Borsuk described a Cantor set in ℝⁿ, whose projection onto any hyperplane contains an (N-1)-dimensional ball, equivalently, has dimension (N-1). In 1994, J.Cobb constructed a Cantor set in ℝ³, whose projection onto any 2-plane is one-dimensional, and posed a general question: does there exist, for given numbers N>m>k>0, a Cantor set in ℝⁿ, whose projection onto any m-plane is of dimension k? (Briefly: (N, m, k)-set.) For the cases (N, m, m-1) and (N, N-1, k) such sets were constructed by O.Frolkina (2010) and S.Barov, J.J.Dijkstra, M.van der Meer (2012), respectively. Different examples for particular cases (N, N-1, N-1) and (N, N-1, N-2) were given by O.Frolkina (2020-21).

    J.Cobb also asked: Cantor sets that raise dimension under all projections and those in general position with respect to all projections are both dense in the Cantor sets in ℝⁿ - which (if either) is more common, in the sense of category or dimension or anything?

    We answer the category part of this question, showing that a generic Cantor set in ℝⁿ has general position with respect to all projections. As a corollary, we obtain our earlier result: all projections of a typical Cantor set are Cantor sets.

Sergei Gukov, California Institute of Technology
Non-semisimple TQFT's and BPS q-series

   Following recent work with Francesco Costantino and Pavel Putrov [arXiv:2107.14238], we will consider various perspectives on the operation of sending the quantum parameter q inside the unit disk to a root of unity. The goal is to see parallels between various manifestations of this limit in quantum groups, in vertex algebras, in exactly solvable lattice models, in the corresponding TQFTs, and in the geometry of Coulomb branches / affine Grassmannians.

Maxim Ivanov, Novosibirsk State University
Recurrent construction of virtual link invariants arising from flat links

    Theory of virtual knots and links was introduced by Kauffman as a generalization of classical knot theory. Flat virtual links are equivalence classes of virtual links with respect to a changing of a type of a classical crossing in a diagram. In 2018 Kaur, Prabhakar and Vesnin introduced a family of invariants of virtual knots, called F-polynomials, based on some invariants of flat virtual links. 
  As a generalization of F-polynomials we present a recurrent construction of invariants of virtual link by using invariants of flat virtual links.  Specifically we define weight functions which assign to every classical crossing in a diagram a value in an abelian group G, satisfying some natural conditions, analogous to Chord Index Axioms. A pair of those weights defines an invariant. We use a recursive procedure to construct various weights and hence, a sequence of invariants for virtual ordered links and flat ordered links. Those invariants appear to be useful in studying connected sums of virtual knots. As an example, we give a new proof of Kishino knot being nontrivial knot.
  This is a joint work with Andrei Vesnin, Amrendra Gill and Madeti Prabhakar (see arxiv:2111.04526).

Suhyeon Jeong, Pusan National University
Pseudoknots and the Goeritz matrix

  In 1933, Goeritz described how a quadratic form could be obtained from a regular projection of a knot, and showed that some of the algebraic invariants of this form are invariants of the knot.
   In 2010, a pseudodiagram was introduced by Ryo Hanaki. A pseudodiagram is a knot or link diagram where we ignore over/under information at some crossings of the diagram.
   In 2012, Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten, and Ljiljana Radvic extended this idea of pseudodiagram to pseudoknots and pseudolinks, i.e. equivalence classes of pseudodiagrams modulo pseudo-Reidemeister moves.
  In this talk, we would like to introduce the Goeritz matrix for a checkerboard colored pseudodiagram of a pseudoknot or pseudolink, which is an extension of the Goeritz matrix for a checkerboard colored diagram of a knot or link. Using this, we show that the determinant, signature, and nullity of oriented classical knots and links extend to those of oriented pseudolinks. This is a joint work with Jieon Kim and Sang Youl Lee.

Louis H Kauffman, University of Illinois at Chicago
Invariants of Flat Virtual Knots and Links

   This talk will discuss the structure of flat virtual knots and links. These correspond to immersions of curves in thickened surfaces taken up to 1-handle stabilization. The theory has a diagrammatic formulation with virtual crossings and flat crossings. Virtual crossings are allowed to make detour moves over all crossings, while flat crossings are restricted to detour only over flat crossings. We will discuss polynomial invariants of flat virtuals and their cobordism properties. We will also discuss what is presently known about reductions of flat virtual diagrams via the combinatorial moves.

Mikhail Khovanov, Columbia University
Introduction to universal construction and foam evaluation

   Foams are two-dimensional cobordisms in 3-space between planar graphs that naturally appear in constructions of link homology theories. We will review foams and their evaluations, that are used in a combinatorial approach to link homology. Foam evaluation utilizes universal construction of topological theories, that give rise to lax TQFTs, where the state space of the union of manifolds properly contains tensor products of their state spaces. 

Jieon Kim, Pusan National University
On symmetric biquandles and invariants for unoriented surface-links

   A quandle is a non-empty set with a binary operation satisfying certain conditions derived from Reidemeister moves. A quandle can be generalized to a biquandle. When invariants are constructed by using quandles and biquandles, orientations are mostly needed. In that case, we only construct invariants for oriented (surface-)links. In 2009, Kamada and Oshiro  constructed symmetric quandles, a quandle with a good involution. It is used to construct invariants for unoriented (surface-)links. In this talk, we introduce symmetric biquandles. By using this algebraic structure, we define invariants for unoriented (surface-)links. 

Seongjeong Kim, Jilin University
On parities for knots in Sg×S¹

   For knots in Sg×S¹, where Sg is an oriented surface of genus g, one of important information is “how many times a half of a crossing turns around , and we call it winding parity of a crossing. In this talk we extend this notion more generally and discuss its geometrical meaning.

Seonhwa Kim, University of Seoul
Generalized Riley polynomials of a knot and trace fields of representations

   We generalize R. Riley's study about parabolic representations of two bridge knots to the general knots in S³. A generalized Riley polynomial Rc(y)∈ℚ[y] is  defined  for any knot diagram with a specified base crossing c,  where the roots also correspond to conjugacy classes of parabolic representations as like the original Riley's.  In particular,  as the sign-type of parabolic quandle is newly introduced, we obtain a formula for the obstruction class to lift a boundary unipotent SL(2,C)-representation and moreover, we can define another polynomial gc(u)∈ ℚ[u], called u-polynomial, and prove that Rc(u²)=±gc(u)gc(-u). Based on this investigation, we introduce Riley field and u-field  closely related to the invariant trace field of representations. Finally we will consider  several open questions  related to them. 

Tatyana Anatolevna Kozlovskaya, Regional Mathematical Center,
Tomsk State University

The singular pure braid group

   We suggest finite set of generators and defining relations for the singular pure braid group SPn. Using this representation, we describe some properties of this group. Also we construct linear representations and representation by automorphisms of free group Fn for the singular braid group SBn.

Sergei Lando, Higher School of Economics, Skolkovo Institute of Science and Technology
Lie algebras weight systems and graph invariants

   Knot invariants are functions on isotopy classes of knots. They are intended to distinguish knots. Vassiliev's theory of finite order knot invariants allows one to associate to each knot invariant a function on chord diagrams
--- simple combinatorial objects consisting of a circle and several chords in it. Such functions are called ``weight systems''. Due to a theorem by Kontsevich, this correspondence is essentially one-to-one: each weight system determines a knot invariant.
  In particular, a weight system can be associated to any semi-simple Lie algebra. It happens, however, that already for the most simple nontrivial case, namely, for the Lie algebra sl2, the computations of the corresponding weight system are very complicated. This case is one of the most important ones because it corresponds to the
famous knot invariant known under the name of colored Jones polynomial.
  The talk will be devoted to known results about computation of Lie algebra weight systems, including recent ones, as well as to open problems in the subject.

Vassily Olegovich Manturov, Moscow Institute of Physics and Technology
Three faces of sliceness for knots: pictures, groups, matchings

   My talk will be devoted to sliceness obstructions for various analogues of knots.  We shall be mostly concerned with the two cases:

  1.  Free knots (capped by formal folded 2-discs).

  2.  Knots in the full torus (to be capped by a disc in ×D³).

   A naive approach requires some "matchings" between crossings to be paired: say, for a 2-component link a mixed crossing can not be paired with a pure crossing. We'll see that one can strongly generalise the "matching" approach.

Sergey Melikhov, Steklov Mathematical Institute
Topological isotopy and Cochran's derived invariants

   We construct a link in the 3-space that is not isotopic to any PL link (non-ambiently). Moreover, there exist uncountably many I-equivalence classes of links in the 3-space.
  The proofs are based on Cochran's derived invariants. We also note a formula expressing Cochran's derived invariants in terms of the Conway polynomial (using bands).
  The details are available in arXiv:2011.01409.

Sam Nelson, Claremont McKenna College
Biquandle Bracket Quivers

   Biquandle brackets are skein invariants for biquandle-colored knots. Taken over the set of biquandle colorings of an oriented knot or link, the multiset of biquandle bracket values is an invariant from which can be recovered the biquandle counting invariant and, depending on the particular biquandle bracket, classical skein invariants as well as biquandle 2-cocycle invariants. A set of biquandle endomorphisms endows the multiset with an invariant quiver structure, proving a categorification of these invariants distinct from the Khovanov-style homological categorifications.

Vladimir Mikhailovich Nezhinskij, St Petersburg State University, Herzen University
Spatial framed graphs and their isotopy

    The problem of isotopy classification of spatial framed graphs equipped with an additional structure - a skeleton, an oriented vertex and a marked point on its  boundary, is reduced to the problem of isotopy classification of tangles. The results are contained in [1] and [2].
     1. V. M. Nezhinskij, Isotopy invariants of spatial graphs, Siberian Electronic Mathematical Reports, 17 (2020), 769 - 776.
     2. V. M. Nezhinskij, Spatial graphs and their isotopy classification, Siberian Electronic Mathematical Reports (manuscript submitted for publication)

Igor Nikonov, Moscow State University
On the principle of indistinguishability for crossings of classical knots

A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show that one can omit the second condition in the case of the universal index. As a consequence, we get the following principle of indistinguishability for classical knots: crossings of the same sign in a classical knot diagram can not be distinguished by any inherent property.

Eiji Ogasa, Meiji Gakuin University
Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links

     We use virtual links, and introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. By using our new invariants, we give a new invariant of classical knots and links in the 3-sphere. Virtual link invariants make new classical knot and link invariants. These invariants are new, nontrivial, and calculable.

Jinseok Oh, Kyungpook National Unversity
Torsion subgroups of homology of finite dihedral quandles

   The free parts of rack and quandle homology were completely determined by Litherland and Nelson, Etingof and Grana independently. However, little is known that the torsion parts are related to the orders of torsion elements in rack homology. For example, it is known that the order of a finite quasigroup quandle annihilates the torsion subgroup of its rack homology. In this talk, we will discuss the torsion subgroups of homology of dihedral quandles.

   This is joint work with Seung Yeop Yang.

Mohd Ibrahim Sheikh, Pusan National University
Bikei’s, Biquandles and Dichormatic Links

   Bikei and Biquandle are algebraic structures whose axioms are motivated by classical Reidemeister moves, and are used to construct invariants for classical knots and links. In this talk we will define their generalizations for dichromatic links and construct invariants for these links.

Hongdae Yun, Kyungpook National Unversity
The Betti numbers for the set-theoretic Yang-Baxter homology groups
of Alexander biquandles

   The Yang-Baxter equation was introduced independently by C. N. Yang(1967) and R. J. Baxter(1972), and it has become important role in the study of knot theory and quantum physics, etc. Biquandles are special families of solutions to the set-theoretic Yang-Baxter equation. A homology theory for the set-theoretic Yang-Baxter equation was introduced by Carter, Elhamdadi, and Saito. In this talk, we first review the definition of set-theoretic Yang-Baxter (co)homology and determine the Betti numbers for the set-theoretic Yang-Baxter (co)homology groups of some finite Alexander biquandles.
  This is joint work with Seung Yeop Yang, Jinseok Oh, Donghan Kim.

Bao Huu Vuong, Tomsk State University
On hyperelliptic Euclidean 3-manifolds

We  study closed orientable Euclidean manifolds which are also known as flat 3-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, R. H. Fox  showed that the 3-torus is not a double branched covering of  the 3-sphere. Then, it is not a hyperelliptic manifold. We show that all the remaining Euclidean 3-forms are hyperelliptic manifolds. This is a joint work with A. D. Mednykh.

Seung Yeop Yang, Kyungpook National Unversity
Set-theoretic Yang-Baxter cohomology groups of finite cyclic biquandles

   Set-theoretic Yang-Baxter (co)homology groups of biquandles and their cocycles can be applied to construct invariants of knots and links. In this talk, we determine the free parts completely and estimate the torsion parts of the integral set-theoretic Yang-Baxter cohomology groups of finite cyclic biquandles. This is joint work with Xiao Wang.